GraphQL API - Query Groups by id

We want to use Group-level Access Tokens to auth against the GraphQL API.
The docs show that Groups can be queried by Query.Group(fullPath: ID!), but can we query them by id? The id is the info that we have immediately available from the access token.
I’m running this query to test how things work:

{
  group(fullPath: "MY-ORG") {
    id
    projects {
      count
      nodes {
        id
        name
      }
    }
  }
  currentUser{
    id
    groupMemberships{
      nodes{
        group{
          id
          fullName
          fullPath
        }
      }
    }
  }
}